Solution

Assume a hydrostatic, vertically well-mixed, inviscid shallow layer, negligible ambient horizontal velocity, and uniform channel width. Volume, mass, and horizontal momentum balances are

h_{t} + (h u)_{x} = w_{e} - w_{d},

(1)

(ρ_{1} h)_{t} + (ρ_{1} h u)_{x} = ρ_{2} w_{e} - ρ_{1} w_{d},

(2)

(ρ_{1} h u)_{t} + (ρ_{1} h u^{2} + \frac{1}{2} g (ρ_{1} - ρ_{2}) h^{2})_{x} = - ρ_{1} u w_{d} .

(3)

Combining the first two gives

h (ρ_{1 t} + u ρ_{1 x}) = (ρ_{2} - ρ_{1}) w_{e} .

(4)

Detrainment removes fluid with the layer's own density and velocity, so it cancels from the corresponding material density and velocity equations; entrainment dilutes and exerts a momentum load.

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Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 345 / 2 / a / Solution

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